Discrete Dynamics in Nature and Society

Volume 2018, Article ID 2903058, 11 pages

https://doi.org/10.1155/2018/2903058

## Group Decision-Making Approach without Weighted Aggregation Operators

College of Economics and Management, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Correspondence should be addressed to Zhiqing Meng; nc.ude.tujz@gniqihzgnem

Received 14 May 2018; Revised 18 June 2018; Accepted 27 June 2018; Published 12 July 2018

Academic Editor: Agnieszka B. Malinowska

Copyright © 2018 Min Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper introduces an approach for group decision-making problems (GDMP) without weighted aggregation operators. This approach is more suitable for scenarios with infinite number of individuals. A mathematical model approach is established based on the new concept of -optimal concession equilibrium solution without weighted aggregation operators for group decision-making problems. It is of practical significance for all decision-makers (experts) to find an optimal solution or to sort out all the candidate solutions. We prove that the -optimal concession equilibrium solution is equivalent to solving a single objective optimization problem, and, under certain conditions, the -optimal equilibrium solution always exists. Moreover, it is proven that the -optimal concession equilibrium solution is equivalent to the robust optimal solution of the group weight aggregation and the optimal solution under the worst weighted aggregation operators.

#### 1. Introduction

The group decision-making problem has always been a hot topic in the research of decision theory and an important branch of scientific research. Many applications are in social network, supplier selection, competitive business environment, economic analysis, strategic planning, medical diagnosis, venture capital, and so forth. The mainstream method of group decision-making is studied through a ranking scheme by a selection function, that is, ranking the preference of individuals in groups. For the scenario with limited number of individuals, this method is very effective. But, for that with the infinite number, it becomes difficult or invalid. For example, the retailer and the manufacturer decide the quantity (in unit of kilogram) of production and supply together in the supply chain such as food (Sharafali and Co, 2000) [1]. The number of productions of the product is infinite.

In the last twenty years, almost all approaches studied are either the weighted aggregation operator methods or the weighted utility methods. There are decision-makers or experts in GDMP, a group of experts (DMs) including , with their cost or benefit evaluating function (scored as a candidate), , for , where is an individual candidate (called a solution) and a set of all individual candidates . Each selects a best individual candidate or sorts out a best one from by evaluating his evaluating function . In general, the group decision method is to establish a group utility function: where is a weight value of ; . All DMs select the best individual candidate or sort out one from by . The different weights of DMs lead to different sorting results. Because usually the DMs are from different fields, the weights of DMs are then different. Therefore, how to take the weights of DMs is very important.

It is found that many researches focus on the weighted aggregation of DMs. For example, Choi (1998) [2] and Kim (1999) [3] et al. presented and developed a mathematical programming model that can establish dominance relations among the preference information about utilities, attribute weights, and group member’s weights. Wei (2000) [4] et al. described a minimax principle based procedure of preference adjustments with a finite number of steps to find the compromise weight. A preemptive goal programming method was proposed for aggregating OWA operator weights (Wang et al., 2007) [5]. Sadi-Nezhad et al. (2008) [6] investigated the generation of a possibilistic model for multidimensional analysis of preference, where the model assesses the fuzzy weights as well as locating the ideal solution with fuzzy decision-making preference on attributes and fuzzy decision matrix. Another good approach aggregating these individual decision matrices into a group decision matrix by using the additive weighted aggregation (AWA) operator was developed by Xu (2009) [7]; then a convergent iterative algorithm to gain a consentaneous group decision matrix is established. And Wu et al. (2009 and 2010) [8, 9] developed some induced continuous ordered weighted geometric (ICOWG) operators and studied some desired properties of the ICOWG operator, where the ordering of the argument values based upon the reliability of the information sources is applied. Yue (2012) [10] determined the weights of decision-makers (DMs), where the weights of decision-makers derived from individual decision are determined with interval numbers.

Some new weight aggregation operator methods have been proposed. For example, Zhou, Chen, and Liu (2012) [11] presented a new aggregation operator called the generalized ordered weighted exponential proportional averaging (GOWEPA) operator, which is based on an optimal model. Liu, Cai, and Martnez (2013) [12] proposed the important weighted continuous generalized ordered weighted averaging (IW-CGOWA) operator and its attitudinal character. Liu, Zhang, and Zhang (2014) [13] studied a group decision-making model based on a generalized ordered weighted geometric average operator with interval preference matrices. Merig, Casanovas, and Yang (2014) [14] introduced the uncertain generalized probabilistic weighted averaging (UGPWA) operator.

Many good methods of multiple attribute group decision-making have been studied and applied to many practical fields. For example, Qi, Liang, and Zhang (2015) [15] focused on the multiple attribute decision-making problems widespread in industry engineering, typically the supplier selection problems, and investigated effective methods utilizing preference information objectively for multiple attribute group decision-making (MAGDM) with unknown attribute weights and expert weights under interval-valued intuitionistic fuzzy environments (IVIFEs). Gao, Li, and Liu (2015) [16] developed a new class of aggregation operator based on utility function, which introduces the risk attitude of decision-makers (DMs) in the aggregation process of investment problem.

In recent years, some new complex methods are put forward for group decision-making in uncertain or fuzzy environment. For example, Yan and Ma (2015) [17] proposed a novel two-stage group decision-making approach to simultaneously address the two types of uncertainties underlying quality function deployment. Dong, Xiao, Zhang, and Wang (2016) [18] put forward a novel consensus framework to manage the consensus and weights (i.e., weights of the experts and attributes) in the iterative multiple attribute group decision-making (MAGDM) problem. Xu, Chen, Rodrguez, Herrera, and Wang (2016) [19] introduced a new type of fuzzy preference structure, called incomplete HFPRs, to describe hesitant and incomplete evaluation information in the group decision-making (GDM) process. In order to eliminate the limitations of deterministic and fuzzy MAGDM methods, Bayrama and Sahin (2016) [20] presented a probabilistic methodology, which is based on TOPSIS and Monte Carlo simulation of triangular data. Chen and Kuo (2017) [21] proposed a new method for autocratic decision-making using group recommendations based on interval type-2 fuzzy sets, enhanced Karnik-Mendel (EKM) algorithms, and the ordered weighted aggregation (OWA) operator. Banaeian, Mobli, Fahimnia, Nielsen, and Omid (2018) [22] compared the application of three popular multicriteria supplier selection methods in the fuzzy environment.

The hesitant fuzzy linguistic term set (HFLTS) and the linguistic distribution are becoming popular tools in modelling linguistic expressions with multiple linguistic terms in decision problems. For example, Wu, Li, Chen, and Dong (2018) [23] proposed a new linguistic group decision model called the maximum support degree model, aiming at maximizing the support degree of the group opinion as well as guarantying the accuracy of the group opinion. Wu, Dai, Chiclana, Fujita, and Herrera-Viedma (2018) [24] presented a minimum adjustment cost feedback mechanism for higher consensus in social network group decision-making under distributed linguistic trust information. Li, Rodrguez, Martnez, Dong, and Herrera (2018) [25] personalized individual semantics in the hesitant GDM with comparative linguistic expressions to show the individual difference in understanding the meaning of words. Furthermore, Wu and Xu (2016) [26] presented a new framework model to address multiple attribute GDM with hesitant fuzzy linguistic information, a good idea where the hesitant fuzzy linguistic weighted average operator and the hesitant fuzzy linguistic ordered weighted average operator are proposed. Wu and Xu (2018) [27] proposed the large-scale group decision-making consensus model in which the clusters are allowed to change and the decision-makers provide preferences using fuzzy preference relations. Wu, Jin, and Xu (2018) [28] developed a new consensus measure that is based on the distances between the individuals on three levels: an alternative pair level, an alternative level, and a preference relation level. Also they designed an algorithm that adopts a local feedback strategy to improve the consensus reaching process.

All the above weighted aggregation methods are more suitable for group decision-making problems with limited number of alternatives. It is not easy to apply the above methods when there is unlimited number of alternatives in product selection of supply chain. Evidently, the cooperation in the supply chain plays an important role in the efficiency of the supply chain (Sharafali and Co, 2000) [1]. Xu, Meng, and Shen (2015) [29] introduced an -optimal equilibrium solution in a model that can decide the minimum concession that the manufacturer and the retailer need to give for their cooperation and proved that the -optimal equilibrium solution can be obtained by solving a goal programming problem. Based on the idea, this paper introduces a new concept of solution to GDMP: -concession equilibrium solution and -optimal concession equilibrium solution. It is characterized by the fact that, for each decision-maker, each objective attribute gives the corresponding concession value , and the -optimal concession equilibrium solution is the minimum concession given. We also prove that any solution to GDMP is an -concession equilibrium solution. It shows that, regardless of the method used to obtain a solution to a group decision, the solution is an -concession equilibrium solution. Therefore, the -optimal concession equilibrium solution provides a natural criterion for evaluating the merits of alternatives. Obviously, it is different from other existing methods with weighted aggregation operators. We explain the rationale of the s-optimal concession equilibrium solution. According to the definition of -optimal concession equilibrium solution, the -optimal concession equilibrium solution is obviously not dependent on the decision function of one DM. Therefore, it is not coercive. That is, the final solutions are not determined by some DMs. On the other hand, for each individual, the equilibrium value is the same for each goal for each decision-maker. Therefore, the -optimal concession equilibrium solution has its individual rationality. The -optimal concession equilibrium solution has the Pareto validity (i.e., the artificial ill solution of all group decision-makers must not be the -optimal concession equilibrium solution).

On the other hand, the linear weighted aggregation method is a very common method used in group decision-making. So, the determination of weights is always the focus of the study in all references. But, its fatal weakness is that different weights lead to different sorting, and it is impossible to prove which linear weighting aggregation method is the best. In this paper, it is proven that the optimal concession equilibrium solution is equivalent to the robust optimal solution of the weighted linear weights of all the decision-makers at the worst weights. When the decision-maker deviates from the worst weight situation, the given weight or the calculated weight is used to compute the optimal solution obtained by the linear weighting method, which is not necessarily the optimal equilibrium solution. Therefore, the motivation and innovation of this paper are that the proposed optimal concession equilibrium solution is a natural law of the consistency of group interests without weighted aggregation operators and when the candidate is infinite, this method avoids the difficulty of calculating the weights so that it is more effective in solving group decision-making problems. In the second section, we discuss the problem of s-optimal concession equilibrium solution with the same concession value. It is proven that the -optimal concession equilibrium solution is equivalent to solving a single objective optimization problem. In the third section, we discuss the problem of a product ordering and production operation decision between the retailer and the manufacturer using the -optimal concession equilibrium solution. Numerical results show that the -optimal concession equilibrium solution increases the quantity of order and production.

#### 2. -Optimal Concession Equilibrium Solution

Suppose that there are decision-makers, , a group of experts and their evaluating (cost or benefit) function , , for . Let be a candidate scheme (solution) and let be a set of all candidate schemes. Each decision-maker selects a solution or sorts out one from by evaluating the function . We have the following group decision-making problem: Let .

*Definition 1. *Let , and . Ifthen is called an -concession equilibrium solution to (GDMP) at the value . is called the target concession value of . is called an equilibrium value of GDMP. The set of all equilibrium values of GDMP is denoted as . If is the -concession equilibrium solution to GDMP and is the minimum of the set , then is called the -optimal concession equilibrium solution to GDMP at the value . is called the optimal equilibrium value, and obviously the optimal equilibrium value is unique. Obviously, we have the following properties.

*Property 2. *Let be the -concession equilibrium solution to GDMP at the value .

(1) Then(2) Ifthen .

(3) Then is the 0-concession equilibrium solution to GDMP at the value , where .

(4) If , then is the -concession equilibrium solution to GDMP at the value** 0**.

(5) If has only a finite number of solutions, then the number of -concession equilibrium solutions is finite, and there must be an -optimal concession equilibrium solution to GDMP.

According to Property 2, the -concession equilibrium solution is to give the corresponding concession value to each of the decision-makers on the decision set or the candidate set . Each decision-maker wants to identify the minimum concession equilibrium value of as the optimal solution for all the decision-makers. Given their respective concession values , the optimal concession equilibrium solution of the minimum equilibrium value is the best solution they all can accept. Therefore, the -optimal concession equilibrium solution can be seen as a fair solution to GDMP.

What interests us is the minimum equilibrium value and its corresponding -optimal concession equilibrium solution in all equilibrium values. We have an example as follows.

*Example 3. *Consider the following GDMP:Obviously, is the optimal solution of decision-maker 1, and is the optimal solution of decision-maker 2.

If , one has is the 2-optimal concession equilibrium solution to the problem at the concession value (0,0). This solution gives the minimum equilibrium value of each decision-maker’s individual objective.

If , one has is the 0-optimal concession equilibrium solution to the problem at the concession value (2,2). This solution gives the minimum equilibrium value of each decision-maker’s individual objective.

If , one has is the 0-optimal concession equilibrium solution to the problem at the concession value (8,0).

The above results show that the concession or compromise values given by the decision-makers are different, and the different optimal concession equilibrium solutions are obtained.

Now, we define the order in the set .

*Definition 4. *Let , and . Let be an -concession equilibrium solution to GDMP at the value and let be an -concession equilibrium solution to GDMP at the value . If , we denote to indicate that is superior to . If , we denote to indicate that is equivalent to .

Obviously, the set is a serially ordered set about the order or .

*Property 5. *Let , and . Let be an -concession equilibrium solution to GDMP at the value and let be an -concession equilibrium solution to GDMP at the value . If , then or .

*Proof. *According to assumption, we haveBy Property 2, we have .

Property 5 shows that the optimal concession equilibrium solution must be nondominated for all decision-makers.

*Example 6. *Consider a GDMP, where three decision-makers try to sort out one from three candidate solutions as scoring in Table 1.